No. All it means is, you've done that Skillshot before so doesn't have to be done again . The easiest way to see which Skillshots you have and haven't done, is to load the final Chapter then bring up the Skllshot list and take note of all the ones that don't have a tick by them .

Holy Potatoes: What the Hell?! can go on the list as well, because that also came out yesterday but the list doesn't sync : https://store.playstation.com/en-gb/product/EP4034-CUSA12062_00-HPWHATTHEHELLEU0

The vast majority of save files are region locked, thus you cannot mix and match where you buy the game from e.g UK saves won't work with an Asian game, US saves won't work with a JP game etc. etc. In your case, your original saves are from an EU copy so won't work with the US release and vice versa. You now have a choice to make, either start over from scratch with your new US copy or fork out even more money to rebuy an EU copy so your original saves will work again.
Some games have retail and digital compatibility issues, even when bought from the same country and it's even possible to have issues with GOTY and non-GOTY Editions of the same game when bought from the same country.

No. It just means if you already have a mission that failed to give you a reward at the end of it, this patch won't fix it, therefore you will have to wait until the next patch to arrive and retroactively give the reward you should have already gotten .

If you have the retail version, then you just reinstall the game without the patch and get the trophy that way but if you have the digital version, you will have to use the downgrade exploit to return it to 1.0.

Thanks anyway, it looks like this is another case of the devs dragging their heels when it comes to syncing content across regions so those other lists probably won't get the respective DLCs for ages yet .
Also, China - the other untagged list is shared between the NA and EU versions : https://psnprofiles.com/trophies/10897-hardcore-mecha

You're going to be waiting a while for the 1.12 patch, because Gearbox have delayed it without giving a new release date : https://www.gamespot.com/articles/borderlands-3-update-delayed-due-to-unrest-over-ra/1100-6478022/
As for a solution to the problem, the all in 1 session trick does work but it's highly dependant on the game not crashing or fouling up in some way .

The Remnant: From the Ashes lists need rescanning please, because a new DLC has just been added . Also, the list for the JP version of Moonlighter - may need rescanning as well since the latest DLC should also be coming out for that as well.

All the DLCs are way easier if you finish Playthrough 2 then return to Playthrough 1, once you are at a much higher level than the bosses and have way more powerful guns. Also, there's a specific order you have to take dragons out in, else they heal and boost each other so become much more powerful but that order escapes me at the moment.